Some noteworthy alternating trilinear forms
type: Item
instance of: scholarly article
title: Some noteworthy alternating trilinear forms
zbMATH Open document ID: 1310.15041
author: Jan Draisma
author: Ron Shaw
published in: Journal of Geometry
publication date: 2014-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.4768
review text: An alternating trilinear form on a vector space \(V\) over a field \(\mathbb F\) is a mapping \(T:V^3\to{\mathbb F}\) which is linear separately in each variable and satisfies \(T(v_1,v_2,v_3)=0\), whenever \(v_i=v_j\) for distinct indices \(i, j\). The authors investigate the existence of alternating trilinear forms with some special properties regarding their zeros. By passing to quotient spaces, one can assume from the start that the form is nondegenerate, i.e., that for each nonzero \(a\in V\) there exist \(b,c\in V\) with \(T(a,b,c)\neq0\). However, \(T\) may also vanish on a projective line, that is, the functional \(T(a,b,\cdot)\) may vanish identically for two distinct points \([a],[b]\) in a projective space \({\mathbb P}V\). In case of finite-dimensional real vector spaces, it is shown that except when \(\dim V\in\{3,7\}\) every alternating trilinear form on \(V\) has this property. The exception at \(\dim V=7\) is related to the existence of eight-dimensional real division algebra of octonions. It is further shown that six-dimensional vector spaces over finite fields admit trilinear alternating forms such that through each point \([a]\in{\mathbb P}V\) there exists a single projective line \([a,b]\subseteq{\mathbb P}V\) with \(T(a,b,\cdot)\) vanishing identically.
reviewed by: Bojan Kuzma
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 51E23
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 17A35
zbMATH DE Number: 6348443
zbMATH Keywords: trivector
zbMATH Keywords: alternating form
zbMATH Keywords: singular line
zbMATH Keywords: Desarguesian line spread
zbMATH Keywords: division algebra
MaRDI profile type: Publication
OpenAlex ID: W2170630483
arXiv ID: 1102.4768
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DOI: 10.1007/S00022-013-0202-2
description: scientific article; zbMATH DE number 6348443